In the analysis of [1]}, it is shown that if \(f\)  is submodular, its associated potential \(g\)  is as well, and this plays a crucial role in the analysis. Here, however, \(f\)  is only weakly submodular, which means we must carry out an alternative analysis to bound the quality of a local optimum for \(g_\phi \) . As in the previous section, we consider two arbitrary bases \(A\)  and \(O\)  of a matroid \(\mathcal {M}\) . As before, we index the elements \(a_i \in A\)  and \(o_i \in O\)  so that \(A - a_i + o_i\)  is a base for all \(1 \le i \le |A|\) . Then, since \(g_{\phi }(A - a_i + o_i) - g(A) = g(o_i | A - a_i) - g(a_i | A - a_i)\) :
\(\sum _{i = 1}^{|A|}g(a_i | A - a_i) = \sum _{i = 1}^{|A|}[g(A) - g(A - a_i + o_i)] + \sum _{i = 1}^{|A|}g(o_i | A - a_i).\) 
